View source for Discrete distribution

{{MSC|60E05}}

[[Category:Distribution theory]]

A probability distribution (concentrated) on a finite or countably infinite set of points of a [[Sampling space|sampling space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330601.png" />. More exactly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330602.png" /> be the sample points and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

be numbers satisfying the conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330604.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Relations (1) and (2) fully define a discrete distribution on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330605.png" />, since the probability measure of any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330606.png" /> is defined by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330607.png" /></td> </tr></table>

Accordingly, the distribution of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330608.png" /> is said to be discrete if it assumes, with probability one, a finite or a countably infinite number of distinct values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d0330609.png" /> with probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306010.png" />. In the case of a distribution on the real line, the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306011.png" /> has jumps at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306012.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306013.png" />, and is constant in the intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306014.png" />. The following discrete distributions occur most frequently: the [[Binomial distribution|binomial distribution]], the [[Geometric distribution|geometric distribution]], the [[Hypergeometric distribution|hypergeometric distribution]], the [[Negative binomial distribution|negative binomial distribution]], the [[Multinomial distribution|multinomial distribution]], and the [[Poisson distribution|Poisson distribution]].



====Comments====
A word of caution. In the Russian literature, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306015.png" />, whereas in Western literature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033060/d03306016.png" />. So the distribution functions are slightly different: left continuous in the Russian literature, and right continuous in the Western literature.